Mathematics

Solve:
$$\int {{{\cos }^{ - 1}}x} \,dx$$


SOLUTION

Consider the given integral.

$$I=\int{{{\cos }^{-1}}xdx}$$

$$I=\int{1.{{\cos }^{-1}}xdx}$$

 

We know that

$$\int{uvdx=u\int{vdx}}-\int{\left( \dfrac{d}{dx}\left( u \right)\int{vdx} \right)}dx$$

 

Therefore,

$$ I={{\cos }^{-1}}x\left( x \right)-\int{\left( -\dfrac{1}{\sqrt{1-{{x}^{2}}}} \right)\left( x \right)}dx $$

$$ I=x{{\cos }^{-1}}x+\int{\dfrac{x}{\sqrt{1-{{x}^{2}}}}}dx $$

 

Let $$t=1-{{x}^{2}}$$

$$ \dfrac{dt}{dx}=0-2x $$

$$ -\dfrac{dt}{2}=xdx $$

 

Therefore,

$$ I=x{{\cos }^{-1}}x-\dfrac{1}{2}\int{\dfrac{dt}{\sqrt{t}}} $$

$$ I=x{{\cos }^{-1}}x-\dfrac{1}{2}\left( 2\sqrt{t} \right)+C $$

$$ I=x{{\cos }^{-1}}x-\sqrt{t}+C $$

 

On putting the value of $$t$$, we get

$$I=x{{\cos }^{-1}}x-\sqrt{1-{{x}^{2}}}+C$$

 

Hence, this is the answer.

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Solve: $$\int _{ -\pi  }^{ \pi  }{ \dfrac { 2x(1+\sin { x } ) }{ 1+\cos ^{ 2 }{ x }  } dx } =$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 One Word Medium
Integrate $$\dfrac{\tan^4 \sqrt x+ \sec^2 \sqrt x}{\sqrt x}$$
The solution is $$\dfrac {2\tan ^3(\sqrt x)}{m}-2\tan \sqrt x+2\sqrt x+2\tan \sqrt x+C$$.Find m

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Solve 
$$\dfrac{1}{2}\int{\dfrac{(-4+2x)}{\sqrt{5-4x+x^{2}}}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
The integral $$\displaystyle \int_{0}^{\pi/4}\frac{\sin^{9}x}{\cos^{11}x}dx=$$
  • A. $$10$$
  • B. $$5$$
  • C. $$\dfrac{1}{5}$$
  • D. $$\dfrac{1}{10}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
$$\displaystyle\int \left(e^x\right)^2 e^x dx$$ is equal to

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer